---
title: "Chapter 3 Beta-Binomial Bayesian Model Notes"
author: "Emanuel Rodriguez"
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---

```{r}
library(bayesrules)
library(tidyverse)
```

The chapter is set up with an example of polling results. We are put into 
the scenario where we are managig the campaing for a candidate. We know 
that on average her support based on recent polls is around 45%. In the 
next few sections we'll work through our Bayesian framework and incorporate
a new tool the **Beta-Binomial** model. This model will take develop a 
continuous prior, as opposed to the discrete one's we've been working with 
so far. 


## The Beta prior

:::{.callout-note}
## Probability Density Function

Let $\pi$ be a continuous random variable with probability density
function (pdf) $f(\pi)$. Then $f(\pi)$ has the following properties:

1. $f(\pi) \geq 0$
2. $\int_{\pi}f(\pi)d\pi = 1$ (this is analogous to $\sum$ in the case of pmfs)
3. $P(a < \pi < b) = \int_a^bf(\pi)d\pi$ when $a\leq b$
:::

:::{.callout-tip icon="true"}
a quick note on (1) above. Note that it does not place a restriction on
$f(\pi)$ being less than 1. This means that we can't interpret values of 
$f$ as probabilities, we can however use to interpret plausability of 
two different events, the greater the value of $f$ the more plausible.
To calculate probabilities using $f$ we must determine the area under the
curve it defines, as shown in (3).  
:::

```{r}

x <- seq(0, 1, by = .05)
y1 <- dbeta(x=x, 5, 5)
y2 <- dbeta(x=x, 5, 1)
y3 <- dbeta(x=x, 1, 5)

d <- tibble(
    x, 
    `beta(5, 1)`=y2, 
    `beta(5, 5)`=y1, 
    `beta(1, 5)`=y3
) |>
    pivot_longer(names_to = "beta_shape", values_to="beta", 
    -x) |>
    mutate(beta_shape=factor(beta_shape, 
                            levels=c("beta(5, 1)",
                                    "beta(5, 5)", 
                                    "beta(1, 5)")))
```

```{r}
#| label: fig-beta-shapes
#| fig-cap: The basic shapes of beta based on the hyperparameters

ggplot(data=d, aes(x, beta)) + geom_point() +
    geom_line() + 
    facet_wrap(vars(beta_shape))
```

In general the shape of the beta distribution is skewed-left when 
$\alpha > \beta$, symmetrical when $\alpha = \beta$ and skewed-right
when $\alpha < \beta$, see @fig-beta-shapes. 

:::{.callout-note}
## The Standard Uniform 

When $\pi$ can take equally take on any value between 0 and 1,
we can model $\pi$ using the standard uniform model.

$$\pi \sim Unif(0, 1)$$

the pdf of $Unif(0, 1)$ is $\f(\pi) = 1$

Note that $Unif(0, 1)$ is just a special case of the Beta with
hyperparameters $\alpha = \beta = 1$, see @fig-std-unif-as-beta
:::

```{r}
#| label: fig-std-unif-as-beta
#| fig-cap: "The standard uniform is a special case of the beta distrubtion with a = b = 1"
std_unif <- tibble(
    x, `beta(1, 1)`=dbeta(x, 1, 1)
)

ggplot(data=std_unif, aes(x, `beta(1, 1)`)) + 
    geom_point() + 
    geom_line()
```

### Mean and Mode of the Beta

The mean and mode are both measures of centrality. The mean is average 
value the mode is the most "common", in the case of pmf this is just
the value that occurs the most in the pdf its the max value. 

The formulations of these for the beta are:

$$E(\pi) = \frac{\alpha}{\alpha + \beta}$$
$$\text{Mode}(\pi) = \frac{\alpha - 1}{\alpha + \beta -2}\;\; \text{when} \;\;\alpha,\beta > 1$$


When can also measure the variability of $\pi$. Take @fig-beta-vars
we can see the variability of $\pi$ differ based on the values 
$\alpha, \beta$.

```{r}
#| label: fig-beta-vars
#| fig-cap: "Two symmetrical shapes of beta with different variance"
beta_variances <- tibble(
    x, 
    `beta(5, 5)`=dbeta(x, 5, 5), 
    `beta(20, 20)`=dbeta(x, 20, 20)
) |>
    pivot_longer(names_to = "beta_shape", values_to = "beta", -x)

ggplot(data=beta_variances, aes(x, beta)) + 
    geom_point() + 
    geom_line() + 
    facet_wrap(vars(beta_shape))
```

We can formulate the variance of $Beta(\alpha, \beta)$ with

$$Var(\pi) = \frac{\alpha\beta}{(\alpha + \beta)^2(\alpha+\beta+1)}$$

it follows that

$$SD = \sqrt{Var(\pi)}$$